This paper focuses on the design of an asynchronous dual solver suitable for model predictive control (MPC) applications. The proposed solver relies on a state-of-the-art variance reduction (VR) scheme, previously used in the context of stochastic proximal gradient methods (Prox-SVRG), and on the alternating minimization algorithm (AMA). The resultant algorithm, a stochastic AMA with VR (SVR-AMA), shows geometric convergence (in the expectation) to a suboptimal solution of the MPC problem and, compared to other state-of-the-art dual asynchronous algorithms, allows one to tune the probability of the asynchronous updates to improve the quality of the estimates. Two novel accelerated versions of the Prox-SVRG (and, by duality, of SVR-AMA) are also provided. We apply the proposed algorithm to a specific class of splitting methods, that is, the decomposition along the length of the prediction horizon. Numerical results on the longitudinal control problem of an Airbus passenger aircraft show the benefits that we can gain in terms of computation time when using our proposed solver with an adaptive probability distribution.@en

The structural evolution, phases and amount of deformation, and the strain distribution of the sedimentary cover of a geologically complex area of 270 km2 in the French subalpine chains (Southeast of France; in the surroundings of the village La Motte-Chalancon) were deduced from the modelling of two folded and faulted competent layers (Tithonian and Barremian formations) both in 2D and 3D. These models were constructed from collected geological data such as BRGM (Bureau de Recherches Geologiques et Minères) geological maps, previous Bachelor’s fieldwork data, new field data and the study of aerial photographs (Google Earth). A grid of 20 vertical 2D cross-sections (10 N-S, and 10 E-W) was generated which was then imported and digitized in the geological modelling software package Move and the 3D model of the competent layers was built, validated and restored to its initial configuration prior to deformation by first removing each fault displacement (Fault Parallel Flow method) and then unfolding each structure (Flexural Slip method) in Move, showing deformation and shortening in two directions: N-S and E-W. The amount of deformation in both directions was estimated from the 3D model. Major deformation took place in the N-S direction with a maximum of 21% shortening for both layers which corresponds to an absolute shortening of 4067m for the Tithonian layer and 4076m for the Barremian layer. Less deformation took place in the E-W direction with a maximum of 9% to 10% for the Tithonian and Barremian horizons respectively which corresponds to an absolute shortening of 1697m and 1976m respectively. This resulted in an area reduction of 16% and 19% of the original area prior to deformation for the Tithonian and Barremian, respectively. Two phases of deformation over geological time were deduced from the orientations of the folds and faults present in the studied area. The presence of NW-SE trending folds and thrust faults (e.g. Mt D’Angele fault, Pommerol fault, or Chalancon fault) along with a conjugate strike-slip system (of one oriented N-S and one NE-SW) reflects the NE-SW compressional stress regime of the first stage of deformation which is related to the Pyrenean phase of the Alpine Orogeny. The second stage, the Alpine phase, which resulted in E-W contractions, is associated with the formation of domal/basinal structures as well as folds with plunging fold axes within the studied area. The northeastern part of the studied area exhibits different fault trends. Two tear faults (L’Aiguille and Ruelles fault), a dextral N-S strike-slip fault (Establet fault), and two E-W trending reverse faults (Peyssias and Hidden fault) resulted from N-S compression. These are related to the first phase of the Pyrenean stage, and then later partly rotated during the latter phase of the Alpine stage. The strain maps produced from the 3D model displaying the high-/low-strain zones of the area mainly show E-W and NW-SE trending strain zones which confirm the direction of the main stress regime of the Pyrenean compressional phase oriented NW-SE.